Lecture - 9 ( calculation of capilarity rise and fall )

Before moving to this lecture I advise you guys to clear your concepts of capillarity which is mention in Lecture - 5. You also need to clear your concepts of Cohesion and adhesion which is mention in Lecture - 1.

Capillarity rise 'h' 

In the figure shown below a beaker is there which is filled with water and a test tube is dip inside it.

Now what going to happen is that we will see Rise of water level inside test tube and it is known as capillarity rise. This rise of water level is denoted by 'h'. The main reason behind this rise water in test tube is adhesion, as we know adhesion is the intermolecular force of attraction between molecules of different nature and here water in contact with glass show adhesion more. But this rise in test tube is up to a certain limit because it is resisted by Gravity. Here to weight of fluid which is acting downward will resist the capillary rise. Here concave meniscus is formed because of adhesion. The reason is that as we know the Weight of fluid is acting on the centroid of the portion of the fluid which is rise in test tube, so from central portion gravity effectively resist the fluid to rise but near the surface of test tube due to the adhesive property of water that is to attract with the molecules of glass there will be a tendency to rise in a test tube which gives us concave meniscus. If cohesive property is dominating then we get a convex meniscus. The figure of concave and convex meniscus is shown below

Let us consider a test tube of glass having diameter 'd' which is dip Inside beaker filled with water as in the figure shown below. 

In red colour the volume of the liquid which rises in test tube is shown separately just to avoid confusion. 'Fs ' represent the surface tension force which is acting on the whole surface of liquid inside test tube. Remember as visible in this figure that surface tension force is acting on two sides, it is incorrect. Surface tension force is acting on the whole surface of liquid. 'θ' is the angle formed between a line tangent to water surface and glass tube. This 'θ' represents angle of contact. This angle of contact will be less than 90 degree because as we know adhesion is dominating here.
This rise in capillarity 'h' is stable only when cosine component of surface tension force will be balanced by weight of fluid which is acting downward. From concepts of  Lecture - 2 we know that weight of fluid (W) is represented as -
W = density*acceleration of gravity*volume
W = ρ*g*V
Therefore weight of fluid which rise in test tube is - 
W = (ρ*g*π*(d^2)*h)/4
For equilibrium condition
Fs cosθ = (ρ*g*π*(d^2)*h)/4
σ*π*d*cosθ = (ρ*g*π*(d^2)*h)/4
In this equation ρ*g = specific weight 'w'
Therefore, σ*π*d*cosθ = (w*π*(d^2)*h)/4
h = (4*σ*cosθ)/(w*d) 
 
From this equation we get one important conclusion that is 'h' is inversely proportional to 'd'
h∝1/d 
It means as we increase the diameter of test tube our h will get decrease. So to avoid the effect of capillarity diameter of test tube should be more. Experimentally it is concluded that to neglect the effect of capillarity the diameter of the tube must be taken greater than 1 cm. 

Capillarity fall 'h'

In the figure shown below the bigger is there which is filled with Mercury and a glass test tube is Dip Inside the Mercury. We will see fall of Mercury level which is represented by 'h'. 
Here capillarity fall occurs and we get a convex shape of meniscus because of cohesive nature of Mercury with respect to glass. As we know Cohesion is the intermolecular forces of attraction between molecules of same nature. 
In the below diagram all the forces are shown. Here 'θ' is the angle made between a line tangent to surface of Mercury and glass test tube. This 'θ' represents angle of contact and it will be greater than 180 degree because Cohesion is dominating. 

This capillarity fall will be stable when cosine component of surface tension force will be in equilibrium with the pressure(P) exerted by Mercury. Here pressure comes in roll because of accumulation of Mercury inside test tube. 
-Fs cosθ = Fp 
Fp represent pressure force 
-σ*π*d*cosθ = P*(π/4)*(d^2)
Pressure of fluid = ρ*g*h
-σ*π*d*cosθ = ρ*g*h*(π/4)*(d^2)
-σ*π*d*cosθ = w*h*(π/4)*(d^2)
h = (-4*σ*cosθ)/(w*d) 




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